Some Practical Guidelines for Effective Sample-Size Determination Russell V. Lenth∗ Department of Statistics University of Iowa March 1, 2001 Abstract Sample-size determination is often an important step in planning a statistical study—and it is usually a difficult one. Among the important hurdles to be surpassed, one must obtain an estimate of one or more error variances, and specify an effect size of importance. There is the temptation to take some shortcuts. This paper offers some suggestions for successful and meaningful sample-size determination. Also discussed is the possibility that sample size may not be the main issue, that the real goal is to design a high-quality study. Finally, criticism is made of some ill-advised shortcuts relating to power and sample size. Key words: Power; Sample size; Observed power; Retrospective power; Study design; Cohen’s effect measures; Equivalence testing; ∗I wish to thank John Castelloe, Kate Cowles, Steve Simon, two referees, an editor, and an associate editor for their helpful comments on earlier drafts of this paper. Much of this work was done with the support of the Obermann Center for Advanced Studies at the University of Iowa. 1 1 Sample size and power Statistical studies (surveys, experiments, observational studies, etc.) are always better when they are care- fully planned. Good planning has many aspects. The problem should be carefully defined and operational- ized. Experimental or observational units must be selected from the appropriate population. The study must be randomized correctly. The procedures must be followed carefully. Reliable instruments should be used to obtain measurements. Finally, the study must be of adequate size, relative to the goals of the study. It must be “big enough” that an effect of such magnitude as to be of scientific significance will also be statistically significant. It is just as important, however, that the study not be “too big,” where an effect of little scientific importance is nevertheless statistically detectable. Sample size is important for economic reasons: An under-sized study can be a waste of resources for not having the capability to produce useful results, while an over-sized one uses more resources than are necessary. In an experiment involving human or animal subjects, sample size is a pivotal issue for ethical reasons. An under-sized experiment exposes the subjects to potentially harmful treatments without advancing knowledge. In an over-sized experiment, an unnecessary number of subjects are exposed to a potentially harmful treatment, or are denied a potentially beneficial one. For such an important issue, there is a surprisingly small amount of published literature. Important gen- eral references include Mace (1964), Kraemer and Thiemann (1987), Cohen (1988), Desu and Raghavarao (1990), Lipsey (1990), Shuster (1990), and Odeh and Fox (1991). There are numerous articles, especially in biostatistics journals, concerning sample-size determination for specific tests. Also of interest are studies of the extent to which sample size is adequate or inadequate in published studies; see Freiman et al. (1986) and Thornley and Adams (1998). There is a growing amount of software for sample-size determination, including nQuery Advisor (Elashoff, 2000), PASS (Hintze, 2000), UnifyPow (O’Brien, 1998), and Power and Precision (Borenstein et al., 1997). Web resources include a comprehensive list of power-analysis soft- ware (Thomas, 1998) and online calculators such as Lenth (2000). Wheeler (1974) provides some useful approximations for use in linear models; Castelloe (2000) gives an up-to-date overview of computational methods. There are several approaches to sample size. For example, one can specify the desired width of a confidence interval and determine the sample size that achieves that goal; or a Bayesian approach can be used where we optimize some utility function—perhaps one that involves both precision of estimation and cost. One of the most popular approaches to sample-size determination involves studying the power of a test of hypothesis. It is the approach emphasized here, although much of the discussion is applicable in other contexts. The power approach involves these elements: 1. Specify a hypothesis test on a parameter θ (along with the underlying probability model for the data). 2. Specify the significance level α of the test. 3. Specify an effect size θ˜ that reflects an alternative of scientific interest. 4. Obtain historical values or estimates of other parameters needed to compute the power function of the test. 5. Specify a target value π˜ of the power of the test when θ = θ˜. Notationally, the power of the test is a function π(θ,n,α,...) where n is the sample size and the “. ” part refers to the additional parameters mentioned in step 4. The required sample size is the smallest integer n such that π(θ˜,n,α,...) ≥ π˜. 2 Figure 1: Software solution (Java applet in Lenth, 2000) to the sample-size problem in the blood-pressure example. Example To illustrate, suppose that we plan to conduct a simple two-sample experiment comparing a treatment with a control. The response variable is systolic blood pressure (SBP), measured using a standard sphygmomanometer. The treatment is supposed to reduce blood pressure; so we set up a one-sided test of H0 : µT = µC versus H1 : µT < µC, where µT is the mean SBP for the treatment group and µC is the mean SBP for the control group. Here, the parameter θ = µT − µC is the effect being tested; thus, in the above framework we would write H0 : θ = 0 and H1 : θ < 0. The goals of the experiment specify that we want to be able to detect a situation where the treatment mean is 15 mm Hg lower than the control group; i.e., the required effect size is θ˜ = −15. We specify that such an effect be detected with 80% power (π˜ = .80) when the significance level is α = .05. Past experience with similar experiments—with similar sphygmomanometers and similar subjects—suggests that the data will be approximately normally distributed with a standard deviation of σ = 20 mm Hg. We plan to use a two-sample pooled t test with equal numbers n of subjects in each group. Now we have all of the specifications needed for determining sample size using the power approach, and their values may be entered in suitable formulas, charts, or power-analysis software. Using the computer dialog shown in Figure 1, we find that a sample size of n = 23 per group is needed to achieve the stated goals. The actual power is .8049. The example shows how the pieces fit together, and that with the help of appropriate software, sample- size determination is not technically difficult. Defining the formal hypotheses and significance level are familiar topics taught in most introductory statistics courses. Deciding on the target power is less familiar. The idea is that we want to have a reasonable chance of detecting the stated effect size. A target value of .80 is fairly common and also somewhat minimal—some authors argue for higher powers such as .85 or .90. As power increases, however, the required sample size increases at an increasing rate. In this example, a target power of π˜ = .95 necessitates a sample size of n = 40—almost 75% more than is needed for a power of .80. The main focus of this article is the remaining specifications (items (3) and (4)). They can present some real difficulties in practice. Who told us that the goal was to detect a mean difference of 15 mm Hg? 3 How do we know that σ = 20, given that we are only planning the experiment and so no data have been collected yet? Such inputs to the sample-size problem are often hard-won, and the purpose of this article is to describe some of the common pitfalls. These pitfalls are fairly well known to practicing statisticians, and are discussed in several applications-oriented papers such as Muller and Benignus (1992) and Thomas (1997); but there is not much discussion of such issues in the “mainstream” statistical literature. Obtaining an effect size of scientific importance requires obtaining meaningful input from the researcher(s) responsible for the study. Conversely, there are technical issues to be addressed that require the expertise of a statistician. Section 2 talks about each of their contributions. Sometimes, there are historical data that can be used to estimate variances and other parameters in the power function. If not, a pilot study is needed. In either case, one must be careful that the data are appropriate. These aspects are discussed in Section 3. In many practical situations, the sample size is mostly or entirely based on non-statistical criteria. Sec- tion 4 offers some suggestions on how to examine such studies and help ensure that they are effective. Section 5 makes the point that not all sample-size problems are the same, nor are they all equally important. It also discusses the interplay between study design and sample size. Since it can be so difficult to address issues such as desired effect size and error variances, people try to bypass them in various ways. One may try to redefine the problem, or rely on arbitrary standards; see Section 6. We also argue against various misguided uses of retrospective power in Section 7. The subsequent exposition makes frequent use of terms such as “science” and “research.” These are intended to be taken very broadly. Such terms refer to the acquisition of knowledge for serious purposes, whether they be advancement of a scholarly discipline, increasing the quality of a manufacturing process, or improving our government’s social services. 2 Eliciting effect size Recall that one step in the sample-size problem requires eliciting an effect size of scientific interest.